Properties

Label 177.8.a.d.1.5
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-13.7979\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.7979 q^{2} +27.0000 q^{3} +35.7870 q^{4} -405.222 q^{5} -345.544 q^{6} -375.715 q^{7} +1180.14 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-12.7979 q^{2} +27.0000 q^{3} +35.7870 q^{4} -405.222 q^{5} -345.544 q^{6} -375.715 q^{7} +1180.14 q^{8} +729.000 q^{9} +5186.00 q^{10} -1868.92 q^{11} +966.249 q^{12} +143.937 q^{13} +4808.37 q^{14} -10941.0 q^{15} -19684.0 q^{16} -7997.65 q^{17} -9329.69 q^{18} -54944.6 q^{19} -14501.7 q^{20} -10144.3 q^{21} +23918.3 q^{22} -57227.2 q^{23} +31863.7 q^{24} +86079.9 q^{25} -1842.09 q^{26} +19683.0 q^{27} -13445.7 q^{28} +44123.3 q^{29} +140022. q^{30} -127609. q^{31} +100857. q^{32} -50460.9 q^{33} +102353. q^{34} +152248. q^{35} +26088.7 q^{36} -343781. q^{37} +703178. q^{38} +3886.30 q^{39} -478217. q^{40} +57315.2 q^{41} +129826. q^{42} +131635. q^{43} -66883.0 q^{44} -295407. q^{45} +732390. q^{46} +13310.0 q^{47} -531469. q^{48} -682381. q^{49} -1.10164e6 q^{50} -215937. q^{51} +5151.07 q^{52} -642722. q^{53} -251902. q^{54} +757328. q^{55} -443395. q^{56} -1.48351e6 q^{57} -564687. q^{58} +205379. q^{59} -391545. q^{60} -1.24234e6 q^{61} +1.63313e6 q^{62} -273896. q^{63} +1.22879e6 q^{64} -58326.4 q^{65} +645794. q^{66} +333473. q^{67} -286212. q^{68} -1.54514e6 q^{69} -1.94846e6 q^{70} +4.40870e6 q^{71} +860319. q^{72} -790210. q^{73} +4.39968e6 q^{74} +2.32416e6 q^{75} -1.96630e6 q^{76} +702181. q^{77} -49736.5 q^{78} +1.80576e6 q^{79} +7.97640e6 q^{80} +531441. q^{81} -733516. q^{82} +3.39335e6 q^{83} -363034. q^{84} +3.24082e6 q^{85} -1.68465e6 q^{86} +1.19133e6 q^{87} -2.20558e6 q^{88} +9.39299e6 q^{89} +3.78060e6 q^{90} -54079.2 q^{91} -2.04799e6 q^{92} -3.44544e6 q^{93} -170341. q^{94} +2.22648e7 q^{95} +2.72315e6 q^{96} +773470. q^{97} +8.73307e6 q^{98} -1.36244e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.7979 −1.13119 −0.565594 0.824684i \(-0.691353\pi\)
−0.565594 + 0.824684i \(0.691353\pi\)
\(3\) 27.0000 0.577350
\(4\) 35.7870 0.279586
\(5\) −405.222 −1.44977 −0.724883 0.688872i \(-0.758106\pi\)
−0.724883 + 0.688872i \(0.758106\pi\)
\(6\) −345.544 −0.653092
\(7\) −375.715 −0.414014 −0.207007 0.978339i \(-0.566372\pi\)
−0.207007 + 0.978339i \(0.566372\pi\)
\(8\) 1180.14 0.814924
\(9\) 729.000 0.333333
\(10\) 5186.00 1.63996
\(11\) −1868.92 −0.423366 −0.211683 0.977338i \(-0.567895\pi\)
−0.211683 + 0.977338i \(0.567895\pi\)
\(12\) 966.249 0.161419
\(13\) 143.937 0.0181706 0.00908532 0.999959i \(-0.497108\pi\)
0.00908532 + 0.999959i \(0.497108\pi\)
\(14\) 4808.37 0.468328
\(15\) −10941.0 −0.837023
\(16\) −19684.0 −1.20142
\(17\) −7997.65 −0.394813 −0.197406 0.980322i \(-0.563252\pi\)
−0.197406 + 0.980322i \(0.563252\pi\)
\(18\) −9329.69 −0.377063
\(19\) −54944.6 −1.83776 −0.918878 0.394543i \(-0.870903\pi\)
−0.918878 + 0.394543i \(0.870903\pi\)
\(20\) −14501.7 −0.405334
\(21\) −10144.3 −0.239031
\(22\) 23918.3 0.478907
\(23\) −57227.2 −0.980743 −0.490372 0.871513i \(-0.663139\pi\)
−0.490372 + 0.871513i \(0.663139\pi\)
\(24\) 31863.7 0.470496
\(25\) 86079.9 1.10182
\(26\) −1842.09 −0.0205544
\(27\) 19683.0 0.192450
\(28\) −13445.7 −0.115753
\(29\) 44123.3 0.335951 0.167975 0.985791i \(-0.446277\pi\)
0.167975 + 0.985791i \(0.446277\pi\)
\(30\) 140022. 0.946830
\(31\) −127609. −0.769333 −0.384667 0.923056i \(-0.625684\pi\)
−0.384667 + 0.923056i \(0.625684\pi\)
\(32\) 100857. 0.544105
\(33\) −50460.9 −0.244431
\(34\) 102353. 0.446607
\(35\) 152248. 0.600224
\(36\) 26088.7 0.0931953
\(37\) −343781. −1.11577 −0.557886 0.829918i \(-0.688387\pi\)
−0.557886 + 0.829918i \(0.688387\pi\)
\(38\) 703178. 2.07885
\(39\) 3886.30 0.0104908
\(40\) −478217. −1.18145
\(41\) 57315.2 0.129875 0.0649376 0.997889i \(-0.479315\pi\)
0.0649376 + 0.997889i \(0.479315\pi\)
\(42\) 129826. 0.270389
\(43\) 131635. 0.252483 0.126241 0.992000i \(-0.459709\pi\)
0.126241 + 0.992000i \(0.459709\pi\)
\(44\) −66883.0 −0.118367
\(45\) −295407. −0.483255
\(46\) 732390. 1.10941
\(47\) 13310.0 0.0186998 0.00934989 0.999956i \(-0.497024\pi\)
0.00934989 + 0.999956i \(0.497024\pi\)
\(48\) −531469. −0.693639
\(49\) −682381. −0.828592
\(50\) −1.10164e6 −1.24637
\(51\) −215937. −0.227945
\(52\) 5151.07 0.00508026
\(53\) −642722. −0.593004 −0.296502 0.955032i \(-0.595820\pi\)
−0.296502 + 0.955032i \(0.595820\pi\)
\(54\) −251902. −0.217697
\(55\) 757328. 0.613782
\(56\) −443395. −0.337390
\(57\) −1.48351e6 −1.06103
\(58\) −564687. −0.380023
\(59\) 205379. 0.130189
\(60\) −391545. −0.234020
\(61\) −1.24234e6 −0.700785 −0.350393 0.936603i \(-0.613952\pi\)
−0.350393 + 0.936603i \(0.613952\pi\)
\(62\) 1.63313e6 0.870261
\(63\) −273896. −0.138005
\(64\) 1.22879e6 0.585932
\(65\) −58326.4 −0.0263432
\(66\) 645794. 0.276497
\(67\) 333473. 0.135456 0.0677282 0.997704i \(-0.478425\pi\)
0.0677282 + 0.997704i \(0.478425\pi\)
\(68\) −286212. −0.110384
\(69\) −1.54514e6 −0.566233
\(70\) −1.94846e6 −0.678966
\(71\) 4.40870e6 1.46186 0.730930 0.682452i \(-0.239086\pi\)
0.730930 + 0.682452i \(0.239086\pi\)
\(72\) 860319. 0.271641
\(73\) −790210. −0.237746 −0.118873 0.992909i \(-0.537928\pi\)
−0.118873 + 0.992909i \(0.537928\pi\)
\(74\) 4.39968e6 1.26215
\(75\) 2.32416e6 0.636137
\(76\) −1.96630e6 −0.513810
\(77\) 702181. 0.175280
\(78\) −49736.5 −0.0118671
\(79\) 1.80576e6 0.412065 0.206032 0.978545i \(-0.433945\pi\)
0.206032 + 0.978545i \(0.433945\pi\)
\(80\) 7.97640e6 1.74177
\(81\) 531441. 0.111111
\(82\) −733516. −0.146913
\(83\) 3.39335e6 0.651411 0.325705 0.945471i \(-0.394398\pi\)
0.325705 + 0.945471i \(0.394398\pi\)
\(84\) −363034. −0.0668297
\(85\) 3.24082e6 0.572386
\(86\) −1.68465e6 −0.285605
\(87\) 1.19133e6 0.193961
\(88\) −2.20558e6 −0.345011
\(89\) 9.39299e6 1.41234 0.706169 0.708043i \(-0.250422\pi\)
0.706169 + 0.708043i \(0.250422\pi\)
\(90\) 3.78060e6 0.546653
\(91\) −54079.2 −0.00752291
\(92\) −2.04799e6 −0.274202
\(93\) −3.44544e6 −0.444175
\(94\) −170341. −0.0211530
\(95\) 2.22648e7 2.66432
\(96\) 2.72315e6 0.314139
\(97\) 773470. 0.0860483 0.0430242 0.999074i \(-0.486301\pi\)
0.0430242 + 0.999074i \(0.486301\pi\)
\(98\) 8.73307e6 0.937293
\(99\) −1.36244e6 −0.141122
\(100\) 3.08054e6 0.308054
\(101\) −654307. −0.0631912 −0.0315956 0.999501i \(-0.510059\pi\)
−0.0315956 + 0.999501i \(0.510059\pi\)
\(102\) 2.76354e6 0.257849
\(103\) 5.35811e6 0.483149 0.241574 0.970382i \(-0.422336\pi\)
0.241574 + 0.970382i \(0.422336\pi\)
\(104\) 169865. 0.0148077
\(105\) 4.11069e6 0.346539
\(106\) 8.22551e6 0.670799
\(107\) −4.95592e6 −0.391094 −0.195547 0.980694i \(-0.562648\pi\)
−0.195547 + 0.980694i \(0.562648\pi\)
\(108\) 704395. 0.0538063
\(109\) 1.10724e7 0.818931 0.409465 0.912326i \(-0.365715\pi\)
0.409465 + 0.912326i \(0.365715\pi\)
\(110\) −9.69223e6 −0.694303
\(111\) −9.28208e6 −0.644191
\(112\) 7.39558e6 0.497404
\(113\) 1.00502e7 0.655239 0.327619 0.944810i \(-0.393754\pi\)
0.327619 + 0.944810i \(0.393754\pi\)
\(114\) 1.89858e7 1.20022
\(115\) 2.31897e7 1.42185
\(116\) 1.57904e6 0.0939270
\(117\) 104930. 0.00605688
\(118\) −2.62843e6 −0.147268
\(119\) 3.00484e6 0.163458
\(120\) −1.29119e7 −0.682110
\(121\) −1.59943e7 −0.820761
\(122\) 1.58993e7 0.792720
\(123\) 1.54751e6 0.0749835
\(124\) −4.56673e6 −0.215095
\(125\) −3.22348e6 −0.147618
\(126\) 3.50530e6 0.156109
\(127\) −2.50572e7 −1.08547 −0.542737 0.839903i \(-0.682612\pi\)
−0.542737 + 0.839903i \(0.682612\pi\)
\(128\) −2.86357e7 −1.20690
\(129\) 3.55414e6 0.145771
\(130\) 746457. 0.0297991
\(131\) −1.29122e7 −0.501823 −0.250912 0.968010i \(-0.580730\pi\)
−0.250912 + 0.968010i \(0.580730\pi\)
\(132\) −1.80584e6 −0.0683394
\(133\) 2.06435e7 0.760857
\(134\) −4.26777e6 −0.153227
\(135\) −7.97598e6 −0.279008
\(136\) −9.43831e6 −0.321742
\(137\) 3.21499e6 0.106821 0.0534106 0.998573i \(-0.482991\pi\)
0.0534106 + 0.998573i \(0.482991\pi\)
\(138\) 1.97745e7 0.640515
\(139\) 1.39325e7 0.440025 0.220012 0.975497i \(-0.429390\pi\)
0.220012 + 0.975497i \(0.429390\pi\)
\(140\) 5.44849e6 0.167814
\(141\) 359371. 0.0107963
\(142\) −5.64222e7 −1.65364
\(143\) −269007. −0.00769284
\(144\) −1.43497e7 −0.400473
\(145\) −1.78797e7 −0.487050
\(146\) 1.01130e7 0.268935
\(147\) −1.84243e7 −0.478388
\(148\) −1.23029e7 −0.311954
\(149\) 4.45947e7 1.10441 0.552206 0.833708i \(-0.313786\pi\)
0.552206 + 0.833708i \(0.313786\pi\)
\(150\) −2.97444e7 −0.719591
\(151\) 3.86098e7 0.912595 0.456298 0.889827i \(-0.349175\pi\)
0.456298 + 0.889827i \(0.349175\pi\)
\(152\) −6.48421e7 −1.49763
\(153\) −5.83029e6 −0.131604
\(154\) −8.98647e6 −0.198274
\(155\) 5.17099e7 1.11535
\(156\) 139079. 0.00293309
\(157\) 6.75271e7 1.39261 0.696305 0.717746i \(-0.254826\pi\)
0.696305 + 0.717746i \(0.254826\pi\)
\(158\) −2.31100e7 −0.466123
\(159\) −1.73535e7 −0.342371
\(160\) −4.08696e7 −0.788825
\(161\) 2.15011e7 0.406042
\(162\) −6.80134e6 −0.125688
\(163\) −3.73402e7 −0.675336 −0.337668 0.941265i \(-0.609638\pi\)
−0.337668 + 0.941265i \(0.609638\pi\)
\(164\) 2.05114e6 0.0363113
\(165\) 2.04478e7 0.354367
\(166\) −4.34278e7 −0.736868
\(167\) 4.08725e7 0.679085 0.339542 0.940591i \(-0.389728\pi\)
0.339542 + 0.940591i \(0.389728\pi\)
\(168\) −1.19717e7 −0.194792
\(169\) −6.27278e7 −0.999670
\(170\) −4.14758e7 −0.647476
\(171\) −4.00546e7 −0.612585
\(172\) 4.71082e6 0.0705906
\(173\) 3.92808e7 0.576791 0.288395 0.957511i \(-0.406878\pi\)
0.288395 + 0.957511i \(0.406878\pi\)
\(174\) −1.52466e7 −0.219406
\(175\) −3.23415e7 −0.456170
\(176\) 3.67879e7 0.508640
\(177\) 5.54523e6 0.0751646
\(178\) −1.20211e8 −1.59762
\(179\) −4.89222e7 −0.637559 −0.318780 0.947829i \(-0.603273\pi\)
−0.318780 + 0.947829i \(0.603273\pi\)
\(180\) −1.05717e7 −0.135111
\(181\) −8.97873e7 −1.12549 −0.562743 0.826632i \(-0.690254\pi\)
−0.562743 + 0.826632i \(0.690254\pi\)
\(182\) 692102. 0.00850982
\(183\) −3.35431e7 −0.404599
\(184\) −6.75359e7 −0.799231
\(185\) 1.39307e8 1.61761
\(186\) 4.40945e7 0.502445
\(187\) 1.49470e7 0.167150
\(188\) 476326. 0.00522819
\(189\) −7.39520e6 −0.0796771
\(190\) −2.84943e8 −3.01384
\(191\) 1.17574e8 1.22094 0.610470 0.792040i \(-0.290981\pi\)
0.610470 + 0.792040i \(0.290981\pi\)
\(192\) 3.31773e7 0.338288
\(193\) −1.89578e8 −1.89818 −0.949092 0.314999i \(-0.897996\pi\)
−0.949092 + 0.314999i \(0.897996\pi\)
\(194\) −9.89881e6 −0.0973368
\(195\) −1.57481e6 −0.0152092
\(196\) −2.44204e7 −0.231663
\(197\) 6.19268e7 0.577094 0.288547 0.957466i \(-0.406828\pi\)
0.288547 + 0.957466i \(0.406828\pi\)
\(198\) 1.74365e7 0.159636
\(199\) 1.68475e8 1.51548 0.757741 0.652556i \(-0.226303\pi\)
0.757741 + 0.652556i \(0.226303\pi\)
\(200\) 1.01586e8 0.897901
\(201\) 9.00378e6 0.0782058
\(202\) 8.37378e6 0.0714812
\(203\) −1.65778e7 −0.139088
\(204\) −7.72772e6 −0.0637303
\(205\) −2.32254e7 −0.188289
\(206\) −6.85727e7 −0.546532
\(207\) −4.17187e7 −0.326914
\(208\) −2.83326e6 −0.0218305
\(209\) 1.02687e8 0.778044
\(210\) −5.26084e7 −0.392001
\(211\) 1.90058e7 0.139283 0.0696413 0.997572i \(-0.477815\pi\)
0.0696413 + 0.997572i \(0.477815\pi\)
\(212\) −2.30011e7 −0.165796
\(213\) 1.19035e8 0.844006
\(214\) 6.34255e7 0.442401
\(215\) −5.33414e7 −0.366041
\(216\) 2.32286e7 0.156832
\(217\) 4.79445e7 0.318515
\(218\) −1.41703e8 −0.926364
\(219\) −2.13357e7 −0.137262
\(220\) 2.71025e7 0.171605
\(221\) −1.15116e6 −0.00717400
\(222\) 1.18791e8 0.728701
\(223\) −9.54195e7 −0.576196 −0.288098 0.957601i \(-0.593023\pi\)
−0.288098 + 0.957601i \(0.593023\pi\)
\(224\) −3.78936e7 −0.225267
\(225\) 6.27522e7 0.367274
\(226\) −1.28622e8 −0.741198
\(227\) 1.24600e7 0.0707012 0.0353506 0.999375i \(-0.488745\pi\)
0.0353506 + 0.999375i \(0.488745\pi\)
\(228\) −5.30902e7 −0.296649
\(229\) −1.55581e8 −0.856118 −0.428059 0.903751i \(-0.640802\pi\)
−0.428059 + 0.903751i \(0.640802\pi\)
\(230\) −2.96781e8 −1.60838
\(231\) 1.89589e7 0.101198
\(232\) 5.20715e7 0.273774
\(233\) −1.59313e8 −0.825100 −0.412550 0.910935i \(-0.635362\pi\)
−0.412550 + 0.910935i \(0.635362\pi\)
\(234\) −1.34289e6 −0.00685147
\(235\) −5.39351e6 −0.0271103
\(236\) 7.34990e6 0.0363990
\(237\) 4.87556e7 0.237906
\(238\) −3.84557e7 −0.184902
\(239\) 2.85477e8 1.35263 0.676314 0.736614i \(-0.263576\pi\)
0.676314 + 0.736614i \(0.263576\pi\)
\(240\) 2.15363e8 1.00561
\(241\) 2.08212e8 0.958177 0.479088 0.877767i \(-0.340967\pi\)
0.479088 + 0.877767i \(0.340967\pi\)
\(242\) 2.04694e8 0.928435
\(243\) 1.43489e7 0.0641500
\(244\) −4.44595e7 −0.195930
\(245\) 2.76516e8 1.20127
\(246\) −1.98049e7 −0.0848204
\(247\) −7.90856e6 −0.0333932
\(248\) −1.50596e8 −0.626948
\(249\) 9.16203e7 0.376092
\(250\) 4.12539e7 0.166984
\(251\) −3.40393e8 −1.35870 −0.679348 0.733816i \(-0.737737\pi\)
−0.679348 + 0.733816i \(0.737737\pi\)
\(252\) −9.80192e6 −0.0385842
\(253\) 1.06953e8 0.415214
\(254\) 3.20680e8 1.22788
\(255\) 8.75022e7 0.330467
\(256\) 2.09193e8 0.779304
\(257\) 1.59818e8 0.587298 0.293649 0.955913i \(-0.405130\pi\)
0.293649 + 0.955913i \(0.405130\pi\)
\(258\) −4.54857e7 −0.164894
\(259\) 1.29164e8 0.461945
\(260\) −2.08733e6 −0.00736518
\(261\) 3.21659e7 0.111984
\(262\) 1.65249e8 0.567656
\(263\) −7.73432e7 −0.262166 −0.131083 0.991371i \(-0.541845\pi\)
−0.131083 + 0.991371i \(0.541845\pi\)
\(264\) −5.95507e7 −0.199192
\(265\) 2.60445e8 0.859717
\(266\) −2.64194e8 −0.860672
\(267\) 2.53611e8 0.815414
\(268\) 1.19340e7 0.0378717
\(269\) −4.47623e8 −1.40210 −0.701050 0.713112i \(-0.747285\pi\)
−0.701050 + 0.713112i \(0.747285\pi\)
\(270\) 1.02076e8 0.315610
\(271\) −3.39469e8 −1.03611 −0.518056 0.855346i \(-0.673344\pi\)
−0.518056 + 0.855346i \(0.673344\pi\)
\(272\) 1.57426e8 0.474335
\(273\) −1.46014e6 −0.00434335
\(274\) −4.11452e7 −0.120835
\(275\) −1.60876e8 −0.466475
\(276\) −5.52958e7 −0.158311
\(277\) −3.30374e8 −0.933957 −0.466979 0.884269i \(-0.654658\pi\)
−0.466979 + 0.884269i \(0.654658\pi\)
\(278\) −1.78307e8 −0.497751
\(279\) −9.30268e7 −0.256444
\(280\) 1.79673e8 0.489137
\(281\) −5.59715e7 −0.150486 −0.0752428 0.997165i \(-0.523973\pi\)
−0.0752428 + 0.997165i \(0.523973\pi\)
\(282\) −4.59920e6 −0.0122127
\(283\) −3.42839e8 −0.899160 −0.449580 0.893240i \(-0.648426\pi\)
−0.449580 + 0.893240i \(0.648426\pi\)
\(284\) 1.57774e8 0.408716
\(285\) 6.01149e8 1.53824
\(286\) 3.44273e6 0.00870205
\(287\) −2.15342e7 −0.0537702
\(288\) 7.35251e7 0.181368
\(289\) −3.46376e8 −0.844123
\(290\) 2.28824e8 0.550945
\(291\) 2.08837e7 0.0496800
\(292\) −2.82792e7 −0.0664703
\(293\) 5.81815e8 1.35129 0.675644 0.737228i \(-0.263865\pi\)
0.675644 + 0.737228i \(0.263865\pi\)
\(294\) 2.35793e8 0.541147
\(295\) −8.32241e7 −0.188743
\(296\) −4.05708e8 −0.909269
\(297\) −3.67860e7 −0.0814769
\(298\) −5.70719e8 −1.24930
\(299\) −8.23711e6 −0.0178207
\(300\) 8.31745e7 0.177855
\(301\) −4.94572e7 −0.104531
\(302\) −4.94125e8 −1.03232
\(303\) −1.76663e7 −0.0364835
\(304\) 1.08153e9 2.20791
\(305\) 5.03422e8 1.01597
\(306\) 7.46156e7 0.148869
\(307\) −6.21817e8 −1.22653 −0.613265 0.789877i \(-0.710144\pi\)
−0.613265 + 0.789877i \(0.710144\pi\)
\(308\) 2.51290e7 0.0490057
\(309\) 1.44669e8 0.278946
\(310\) −6.61779e8 −1.26167
\(311\) −7.86009e8 −1.48172 −0.740860 0.671659i \(-0.765582\pi\)
−0.740860 + 0.671659i \(0.765582\pi\)
\(312\) 4.58636e6 0.00854922
\(313\) −3.72650e8 −0.686904 −0.343452 0.939170i \(-0.611596\pi\)
−0.343452 + 0.939170i \(0.611596\pi\)
\(314\) −8.64207e8 −1.57530
\(315\) 1.10989e8 0.200075
\(316\) 6.46228e7 0.115208
\(317\) 2.38136e8 0.419873 0.209936 0.977715i \(-0.432674\pi\)
0.209936 + 0.977715i \(0.432674\pi\)
\(318\) 2.22089e8 0.387286
\(319\) −8.24630e7 −0.142230
\(320\) −4.97932e8 −0.849465
\(321\) −1.33810e8 −0.225798
\(322\) −2.75170e8 −0.459309
\(323\) 4.39428e8 0.725569
\(324\) 1.90187e7 0.0310651
\(325\) 1.23901e7 0.0200208
\(326\) 4.77877e8 0.763932
\(327\) 2.98953e8 0.472810
\(328\) 6.76397e7 0.105838
\(329\) −5.00077e6 −0.00774197
\(330\) −2.61690e8 −0.400856
\(331\) 6.49978e8 0.985147 0.492573 0.870271i \(-0.336056\pi\)
0.492573 + 0.870271i \(0.336056\pi\)
\(332\) 1.21438e8 0.182125
\(333\) −2.50616e8 −0.371924
\(334\) −5.23084e8 −0.768172
\(335\) −1.35131e8 −0.196380
\(336\) 1.99681e8 0.287176
\(337\) 6.65064e8 0.946583 0.473292 0.880906i \(-0.343066\pi\)
0.473292 + 0.880906i \(0.343066\pi\)
\(338\) 8.02786e8 1.13081
\(339\) 2.71355e8 0.378302
\(340\) 1.15979e8 0.160031
\(341\) 2.38491e8 0.325710
\(342\) 5.12616e8 0.692949
\(343\) 5.65798e8 0.757063
\(344\) 1.55347e8 0.205754
\(345\) 6.26123e8 0.820905
\(346\) −5.02712e8 −0.652459
\(347\) 2.66064e8 0.341848 0.170924 0.985284i \(-0.445325\pi\)
0.170924 + 0.985284i \(0.445325\pi\)
\(348\) 4.26341e7 0.0542288
\(349\) 1.20795e9 1.52111 0.760556 0.649273i \(-0.224927\pi\)
0.760556 + 0.649273i \(0.224927\pi\)
\(350\) 4.13904e8 0.516014
\(351\) 2.83311e6 0.00349694
\(352\) −1.88494e8 −0.230356
\(353\) 2.38306e8 0.288352 0.144176 0.989552i \(-0.453947\pi\)
0.144176 + 0.989552i \(0.453947\pi\)
\(354\) −7.09675e7 −0.0850253
\(355\) −1.78650e9 −2.11936
\(356\) 3.36147e8 0.394870
\(357\) 8.11306e7 0.0943726
\(358\) 6.26103e8 0.721199
\(359\) −1.03023e9 −1.17518 −0.587588 0.809160i \(-0.699922\pi\)
−0.587588 + 0.809160i \(0.699922\pi\)
\(360\) −3.48620e8 −0.393816
\(361\) 2.12504e9 2.37734
\(362\) 1.14909e9 1.27314
\(363\) −4.31846e8 −0.473866
\(364\) −1.93533e6 −0.00210330
\(365\) 3.20210e8 0.344675
\(366\) 4.29282e8 0.457677
\(367\) 1.50357e9 1.58779 0.793893 0.608058i \(-0.208051\pi\)
0.793893 + 0.608058i \(0.208051\pi\)
\(368\) 1.12646e9 1.17828
\(369\) 4.17828e7 0.0432917
\(370\) −1.78285e9 −1.82982
\(371\) 2.41480e8 0.245512
\(372\) −1.23302e8 −0.124185
\(373\) −5.53195e8 −0.551947 −0.275973 0.961165i \(-0.589000\pi\)
−0.275973 + 0.961165i \(0.589000\pi\)
\(374\) −1.91290e8 −0.189079
\(375\) −8.70340e7 −0.0852274
\(376\) 1.57076e7 0.0152389
\(377\) 6.35098e6 0.00610444
\(378\) 9.46432e7 0.0901297
\(379\) 1.05631e9 0.996680 0.498340 0.866982i \(-0.333943\pi\)
0.498340 + 0.866982i \(0.333943\pi\)
\(380\) 7.96789e8 0.744905
\(381\) −6.76544e8 −0.626699
\(382\) −1.50470e9 −1.38111
\(383\) −1.04319e8 −0.0948782 −0.0474391 0.998874i \(-0.515106\pi\)
−0.0474391 + 0.998874i \(0.515106\pi\)
\(384\) −7.73164e8 −0.696807
\(385\) −2.84539e8 −0.254115
\(386\) 2.42621e9 2.14720
\(387\) 9.59618e7 0.0841609
\(388\) 2.76802e7 0.0240579
\(389\) −1.52386e9 −1.31257 −0.656284 0.754514i \(-0.727873\pi\)
−0.656284 + 0.754514i \(0.727873\pi\)
\(390\) 2.01543e7 0.0172045
\(391\) 4.57683e8 0.387210
\(392\) −8.05302e8 −0.675239
\(393\) −3.48629e8 −0.289728
\(394\) −7.92534e8 −0.652802
\(395\) −7.31734e8 −0.597398
\(396\) −4.87577e7 −0.0394558
\(397\) −2.77621e8 −0.222683 −0.111341 0.993782i \(-0.535515\pi\)
−0.111341 + 0.993782i \(0.535515\pi\)
\(398\) −2.15614e9 −1.71429
\(399\) 5.57375e8 0.439281
\(400\) −1.69440e9 −1.32375
\(401\) 3.39521e8 0.262943 0.131471 0.991320i \(-0.458030\pi\)
0.131471 + 0.991320i \(0.458030\pi\)
\(402\) −1.15230e8 −0.0884654
\(403\) −1.83676e7 −0.0139793
\(404\) −2.34157e7 −0.0176674
\(405\) −2.15352e8 −0.161085
\(406\) 2.12161e8 0.157335
\(407\) 6.42499e8 0.472380
\(408\) −2.54834e8 −0.185758
\(409\) 2.23733e9 1.61696 0.808479 0.588525i \(-0.200291\pi\)
0.808479 + 0.588525i \(0.200291\pi\)
\(410\) 2.97237e8 0.212990
\(411\) 8.68047e7 0.0616733
\(412\) 1.91750e8 0.135082
\(413\) −7.71639e7 −0.0539001
\(414\) 5.33912e8 0.369802
\(415\) −1.37506e9 −0.944393
\(416\) 1.45171e7 0.00988674
\(417\) 3.76178e8 0.254049
\(418\) −1.31418e9 −0.880114
\(419\) 1.72385e9 1.14485 0.572427 0.819955i \(-0.306002\pi\)
0.572427 + 0.819955i \(0.306002\pi\)
\(420\) 1.47109e8 0.0968875
\(421\) 2.37385e9 1.55048 0.775239 0.631668i \(-0.217629\pi\)
0.775239 + 0.631668i \(0.217629\pi\)
\(422\) −2.43234e8 −0.157555
\(423\) 9.70301e6 0.00623326
\(424\) −7.58499e8 −0.483253
\(425\) −6.88436e8 −0.435013
\(426\) −1.52340e9 −0.954729
\(427\) 4.66765e8 0.290135
\(428\) −1.77357e8 −0.109344
\(429\) −7.26318e6 −0.00444147
\(430\) 6.82659e8 0.414061
\(431\) −2.92636e9 −1.76059 −0.880294 0.474428i \(-0.842655\pi\)
−0.880294 + 0.474428i \(0.842655\pi\)
\(432\) −3.87441e8 −0.231213
\(433\) −6.83606e8 −0.404668 −0.202334 0.979317i \(-0.564853\pi\)
−0.202334 + 0.979317i \(0.564853\pi\)
\(434\) −6.13591e8 −0.360300
\(435\) −4.82753e8 −0.281198
\(436\) 3.96246e8 0.228961
\(437\) 3.14433e9 1.80237
\(438\) 2.73052e8 0.155270
\(439\) −9.63693e8 −0.543642 −0.271821 0.962348i \(-0.587626\pi\)
−0.271821 + 0.962348i \(0.587626\pi\)
\(440\) 8.93749e8 0.500186
\(441\) −4.97456e8 −0.276197
\(442\) 1.47324e7 0.00811514
\(443\) −1.22895e9 −0.671616 −0.335808 0.941930i \(-0.609009\pi\)
−0.335808 + 0.941930i \(0.609009\pi\)
\(444\) −3.32178e8 −0.180107
\(445\) −3.80624e9 −2.04756
\(446\) 1.22117e9 0.651786
\(447\) 1.20406e9 0.637632
\(448\) −4.61674e8 −0.242584
\(449\) 2.87288e9 1.49781 0.748903 0.662679i \(-0.230581\pi\)
0.748903 + 0.662679i \(0.230581\pi\)
\(450\) −8.03098e8 −0.415456
\(451\) −1.07118e8 −0.0549848
\(452\) 3.59666e8 0.183196
\(453\) 1.04246e9 0.526887
\(454\) −1.59462e8 −0.0799764
\(455\) 2.19141e7 0.0109065
\(456\) −1.75074e9 −0.864657
\(457\) 5.34428e8 0.261929 0.130964 0.991387i \(-0.458193\pi\)
0.130964 + 0.991387i \(0.458193\pi\)
\(458\) 1.99112e9 0.968430
\(459\) −1.57418e8 −0.0759817
\(460\) 8.29891e8 0.397529
\(461\) −2.83200e9 −1.34629 −0.673146 0.739510i \(-0.735057\pi\)
−0.673146 + 0.739510i \(0.735057\pi\)
\(462\) −2.42635e8 −0.114474
\(463\) −2.92579e9 −1.36997 −0.684984 0.728558i \(-0.740191\pi\)
−0.684984 + 0.728558i \(0.740191\pi\)
\(464\) −8.68525e8 −0.403617
\(465\) 1.39617e9 0.643950
\(466\) 2.03888e9 0.933343
\(467\) −1.22471e9 −0.556449 −0.278225 0.960516i \(-0.589746\pi\)
−0.278225 + 0.960516i \(0.589746\pi\)
\(468\) 3.75513e6 0.00169342
\(469\) −1.25291e8 −0.0560809
\(470\) 6.90258e7 0.0306668
\(471\) 1.82323e9 0.804023
\(472\) 2.42375e8 0.106094
\(473\) −2.46015e8 −0.106893
\(474\) −6.23970e8 −0.269116
\(475\) −4.72963e9 −2.02488
\(476\) 1.07534e8 0.0457006
\(477\) −4.68544e8 −0.197668
\(478\) −3.65351e9 −1.53008
\(479\) −1.72272e9 −0.716211 −0.358106 0.933681i \(-0.616577\pi\)
−0.358106 + 0.933681i \(0.616577\pi\)
\(480\) −1.10348e9 −0.455429
\(481\) −4.94827e7 −0.0202743
\(482\) −2.66468e9 −1.08388
\(483\) 5.80530e8 0.234428
\(484\) −5.72388e8 −0.229473
\(485\) −3.13427e8 −0.124750
\(486\) −1.83636e8 −0.0725657
\(487\) 2.39061e9 0.937901 0.468950 0.883225i \(-0.344632\pi\)
0.468950 + 0.883225i \(0.344632\pi\)
\(488\) −1.46613e9 −0.571087
\(489\) −1.00818e9 −0.389905
\(490\) −3.53883e9 −1.35886
\(491\) 5.12479e8 0.195385 0.0976924 0.995217i \(-0.468854\pi\)
0.0976924 + 0.995217i \(0.468854\pi\)
\(492\) 5.53807e7 0.0209643
\(493\) −3.52883e8 −0.132638
\(494\) 1.01213e8 0.0377740
\(495\) 5.52092e8 0.204594
\(496\) 2.51185e9 0.924291
\(497\) −1.65641e9 −0.605231
\(498\) −1.17255e9 −0.425431
\(499\) 3.61805e9 1.30354 0.651768 0.758418i \(-0.274028\pi\)
0.651768 + 0.758418i \(0.274028\pi\)
\(500\) −1.15359e8 −0.0412720
\(501\) 1.10356e9 0.392070
\(502\) 4.35632e9 1.53694
\(503\) 3.85817e9 1.35174 0.675870 0.737021i \(-0.263768\pi\)
0.675870 + 0.737021i \(0.263768\pi\)
\(504\) −3.23235e8 −0.112463
\(505\) 2.65140e8 0.0916125
\(506\) −1.36878e9 −0.469685
\(507\) −1.69365e9 −0.577160
\(508\) −8.96722e8 −0.303483
\(509\) 5.17337e9 1.73885 0.869423 0.494068i \(-0.164491\pi\)
0.869423 + 0.494068i \(0.164491\pi\)
\(510\) −1.11985e9 −0.373821
\(511\) 2.96894e8 0.0984300
\(512\) 9.88137e8 0.325366
\(513\) −1.08148e9 −0.353676
\(514\) −2.04533e9 −0.664345
\(515\) −2.17122e9 −0.700453
\(516\) 1.27192e8 0.0407555
\(517\) −2.48754e7 −0.00791686
\(518\) −1.65303e9 −0.522547
\(519\) 1.06058e9 0.333010
\(520\) −6.88331e7 −0.0214677
\(521\) −5.01575e9 −1.55383 −0.776916 0.629604i \(-0.783217\pi\)
−0.776916 + 0.629604i \(0.783217\pi\)
\(522\) −4.11657e8 −0.126674
\(523\) 3.72739e9 1.13933 0.569664 0.821878i \(-0.307073\pi\)
0.569664 + 0.821878i \(0.307073\pi\)
\(524\) −4.62089e8 −0.140303
\(525\) −8.73220e8 −0.263370
\(526\) 9.89832e8 0.296559
\(527\) 1.02057e9 0.303743
\(528\) 9.93273e8 0.293663
\(529\) −1.29868e8 −0.0381422
\(530\) −3.33316e9 −0.972501
\(531\) 1.49721e8 0.0433963
\(532\) 7.38769e8 0.212725
\(533\) 8.24977e6 0.00235992
\(534\) −3.24569e9 −0.922386
\(535\) 2.00825e9 0.566995
\(536\) 3.93544e8 0.110387
\(537\) −1.32090e9 −0.368095
\(538\) 5.72864e9 1.58604
\(539\) 1.27532e9 0.350798
\(540\) −2.85436e8 −0.0780066
\(541\) −7.61357e8 −0.206727 −0.103364 0.994644i \(-0.532961\pi\)
−0.103364 + 0.994644i \(0.532961\pi\)
\(542\) 4.34449e9 1.17204
\(543\) −2.42426e9 −0.649799
\(544\) −8.06622e8 −0.214820
\(545\) −4.48676e9 −1.18726
\(546\) 1.86868e7 0.00491315
\(547\) −1.94954e9 −0.509304 −0.254652 0.967033i \(-0.581961\pi\)
−0.254652 + 0.967033i \(0.581961\pi\)
\(548\) 1.15055e8 0.0298657
\(549\) −9.05664e8 −0.233595
\(550\) 2.05888e9 0.527670
\(551\) −2.42434e9 −0.617395
\(552\) −1.82347e9 −0.461436
\(553\) −6.78452e8 −0.170601
\(554\) 4.22811e9 1.05648
\(555\) 3.76130e9 0.933927
\(556\) 4.98602e8 0.123025
\(557\) −4.11925e9 −1.01001 −0.505004 0.863117i \(-0.668509\pi\)
−0.505004 + 0.863117i \(0.668509\pi\)
\(558\) 1.19055e9 0.290087
\(559\) 1.89471e7 0.00458777
\(560\) −2.99685e9 −0.721119
\(561\) 4.03568e8 0.0965044
\(562\) 7.16319e8 0.170227
\(563\) −1.25233e9 −0.295760 −0.147880 0.989005i \(-0.547245\pi\)
−0.147880 + 0.989005i \(0.547245\pi\)
\(564\) 1.28608e7 0.00301850
\(565\) −4.07256e9 −0.949943
\(566\) 4.38762e9 1.01712
\(567\) −1.99670e8 −0.0460016
\(568\) 5.20286e9 1.19130
\(569\) −1.82236e9 −0.414708 −0.207354 0.978266i \(-0.566485\pi\)
−0.207354 + 0.978266i \(0.566485\pi\)
\(570\) −7.69346e9 −1.74004
\(571\) −2.71123e9 −0.609453 −0.304727 0.952440i \(-0.598565\pi\)
−0.304727 + 0.952440i \(0.598565\pi\)
\(572\) −9.62694e6 −0.00215081
\(573\) 3.17450e9 0.704910
\(574\) 2.75593e8 0.0608242
\(575\) −4.92611e9 −1.08060
\(576\) 8.95787e8 0.195311
\(577\) −6.72315e8 −0.145699 −0.0728496 0.997343i \(-0.523209\pi\)
−0.0728496 + 0.997343i \(0.523209\pi\)
\(578\) 4.43290e9 0.954862
\(579\) −5.11862e9 −1.09592
\(580\) −6.39862e8 −0.136172
\(581\) −1.27493e9 −0.269693
\(582\) −2.67268e8 −0.0561974
\(583\) 1.20120e9 0.251058
\(584\) −9.32555e8 −0.193744
\(585\) −4.25199e7 −0.00878106
\(586\) −7.44603e9 −1.52856
\(587\) −2.63538e9 −0.537786 −0.268893 0.963170i \(-0.586658\pi\)
−0.268893 + 0.963170i \(0.586658\pi\)
\(588\) −6.59350e8 −0.133750
\(589\) 7.01142e9 1.41385
\(590\) 1.06510e9 0.213504
\(591\) 1.67202e9 0.333185
\(592\) 6.76699e9 1.34051
\(593\) 1.67193e9 0.329251 0.164625 0.986356i \(-0.447359\pi\)
0.164625 + 0.986356i \(0.447359\pi\)
\(594\) 4.70784e8 0.0921657
\(595\) −1.21763e9 −0.236976
\(596\) 1.59591e9 0.308778
\(597\) 4.54883e9 0.874963
\(598\) 1.05418e8 0.0201586
\(599\) 1.58648e9 0.301607 0.150803 0.988564i \(-0.451814\pi\)
0.150803 + 0.988564i \(0.451814\pi\)
\(600\) 2.74282e9 0.518403
\(601\) 1.06586e9 0.200280 0.100140 0.994973i \(-0.468071\pi\)
0.100140 + 0.994973i \(0.468071\pi\)
\(602\) 6.32950e8 0.118245
\(603\) 2.43102e8 0.0451521
\(604\) 1.38173e9 0.255149
\(605\) 6.48124e9 1.18991
\(606\) 2.26092e8 0.0412697
\(607\) 9.08916e9 1.64954 0.824771 0.565467i \(-0.191304\pi\)
0.824771 + 0.565467i \(0.191304\pi\)
\(608\) −5.54157e9 −0.999932
\(609\) −4.47600e8 −0.0803027
\(610\) −6.44276e9 −1.14926
\(611\) 1.91580e6 0.000339787 0
\(612\) −2.08648e8 −0.0367947
\(613\) −3.73946e9 −0.655688 −0.327844 0.944732i \(-0.606322\pi\)
−0.327844 + 0.944732i \(0.606322\pi\)
\(614\) 7.95797e9 1.38744
\(615\) −6.27085e8 −0.108709
\(616\) 8.28669e8 0.142840
\(617\) 5.49157e9 0.941236 0.470618 0.882337i \(-0.344031\pi\)
0.470618 + 0.882337i \(0.344031\pi\)
\(618\) −1.85146e9 −0.315540
\(619\) 6.71900e9 1.13864 0.569321 0.822115i \(-0.307206\pi\)
0.569321 + 0.822115i \(0.307206\pi\)
\(620\) 1.85054e9 0.311837
\(621\) −1.12640e9 −0.188744
\(622\) 1.00593e10 1.67610
\(623\) −3.52908e9 −0.584728
\(624\) −7.64980e7 −0.0126039
\(625\) −5.41876e9 −0.887810
\(626\) 4.76915e9 0.777018
\(627\) 2.77255e9 0.449204
\(628\) 2.41659e9 0.389354
\(629\) 2.74944e9 0.440521
\(630\) −1.42043e9 −0.226322
\(631\) 3.57282e9 0.566119 0.283060 0.959102i \(-0.408651\pi\)
0.283060 + 0.959102i \(0.408651\pi\)
\(632\) 2.13104e9 0.335801
\(633\) 5.13155e8 0.0804148
\(634\) −3.04765e9 −0.474955
\(635\) 1.01537e10 1.57368
\(636\) −6.21029e8 −0.0957221
\(637\) −9.82199e7 −0.0150561
\(638\) 1.05536e9 0.160889
\(639\) 3.21394e9 0.487287
\(640\) 1.16038e10 1.74973
\(641\) −9.37478e8 −0.140591 −0.0702956 0.997526i \(-0.522394\pi\)
−0.0702956 + 0.997526i \(0.522394\pi\)
\(642\) 1.71249e9 0.255420
\(643\) 8.40132e9 1.24626 0.623131 0.782118i \(-0.285860\pi\)
0.623131 + 0.782118i \(0.285860\pi\)
\(644\) 7.69461e8 0.113524
\(645\) −1.44022e9 −0.211334
\(646\) −5.62377e9 −0.820755
\(647\) 1.33915e10 1.94386 0.971931 0.235264i \(-0.0755956\pi\)
0.971931 + 0.235264i \(0.0755956\pi\)
\(648\) 6.27172e8 0.0905471
\(649\) −3.83837e8 −0.0551176
\(650\) −1.58567e8 −0.0226473
\(651\) 1.29450e9 0.183895
\(652\) −1.33629e9 −0.188814
\(653\) −7.85725e9 −1.10427 −0.552134 0.833756i \(-0.686186\pi\)
−0.552134 + 0.833756i \(0.686186\pi\)
\(654\) −3.82599e9 −0.534837
\(655\) 5.23231e9 0.727526
\(656\) −1.12819e9 −0.156034
\(657\) −5.76063e8 −0.0792485
\(658\) 6.39995e7 0.00875762
\(659\) 3.05094e9 0.415273 0.207637 0.978206i \(-0.433423\pi\)
0.207637 + 0.978206i \(0.433423\pi\)
\(660\) 7.31767e8 0.0990761
\(661\) 1.41136e10 1.90079 0.950395 0.311045i \(-0.100679\pi\)
0.950395 + 0.311045i \(0.100679\pi\)
\(662\) −8.31838e9 −1.11439
\(663\) −3.10812e7 −0.00414191
\(664\) 4.00461e9 0.530850
\(665\) −8.36521e9 −1.10306
\(666\) 3.20737e9 0.420716
\(667\) −2.52506e9 −0.329481
\(668\) 1.46270e9 0.189862
\(669\) −2.57633e9 −0.332667
\(670\) 1.72939e9 0.222143
\(671\) 2.32183e9 0.296689
\(672\) −1.02313e9 −0.130058
\(673\) 1.56465e10 1.97863 0.989313 0.145809i \(-0.0465786\pi\)
0.989313 + 0.145809i \(0.0465786\pi\)
\(674\) −8.51144e9 −1.07076
\(675\) 1.69431e9 0.212046
\(676\) −2.24484e9 −0.279494
\(677\) 5.00729e9 0.620215 0.310107 0.950702i \(-0.399635\pi\)
0.310107 + 0.950702i \(0.399635\pi\)
\(678\) −3.47278e9 −0.427931
\(679\) −2.90604e8 −0.0356252
\(680\) 3.82461e9 0.466451
\(681\) 3.36420e8 0.0408194
\(682\) −3.05219e9 −0.368439
\(683\) −1.48727e10 −1.78615 −0.893073 0.449911i \(-0.851456\pi\)
−0.893073 + 0.449911i \(0.851456\pi\)
\(684\) −1.43343e9 −0.171270
\(685\) −1.30278e9 −0.154866
\(686\) −7.24105e9 −0.856381
\(687\) −4.20070e9 −0.494280
\(688\) −2.59110e9 −0.303337
\(689\) −9.25114e7 −0.0107753
\(690\) −8.01308e9 −0.928597
\(691\) −1.34895e8 −0.0155532 −0.00777662 0.999970i \(-0.502475\pi\)
−0.00777662 + 0.999970i \(0.502475\pi\)
\(692\) 1.40574e9 0.161263
\(693\) 5.11890e8 0.0584266
\(694\) −3.40507e9 −0.386695
\(695\) −5.64576e9 −0.637933
\(696\) 1.40593e9 0.158064
\(697\) −4.58387e8 −0.0512764
\(698\) −1.54593e10 −1.72066
\(699\) −4.30146e9 −0.476372
\(700\) −1.15740e9 −0.127539
\(701\) −8.34830e9 −0.915345 −0.457672 0.889121i \(-0.651317\pi\)
−0.457672 + 0.889121i \(0.651317\pi\)
\(702\) −3.62579e7 −0.00395570
\(703\) 1.88889e10 2.05052
\(704\) −2.29651e9 −0.248064
\(705\) −1.45625e8 −0.0156521
\(706\) −3.04982e9 −0.326180
\(707\) 2.45833e8 0.0261621
\(708\) 1.98447e8 0.0210150
\(709\) −1.42535e9 −0.150197 −0.0750983 0.997176i \(-0.523927\pi\)
−0.0750983 + 0.997176i \(0.523927\pi\)
\(710\) 2.28635e10 2.39739
\(711\) 1.31640e9 0.137355
\(712\) 1.10850e10 1.15095
\(713\) 7.30270e9 0.754519
\(714\) −1.03830e9 −0.106753
\(715\) 1.09007e8 0.0111528
\(716\) −1.75078e9 −0.178253
\(717\) 7.70788e9 0.780940
\(718\) 1.31848e10 1.32935
\(719\) −5.22232e9 −0.523978 −0.261989 0.965071i \(-0.584378\pi\)
−0.261989 + 0.965071i \(0.584378\pi\)
\(720\) 5.81480e9 0.580592
\(721\) −2.01312e9 −0.200031
\(722\) −2.71961e10 −2.68922
\(723\) 5.62172e9 0.553204
\(724\) −3.21322e9 −0.314670
\(725\) 3.79813e9 0.370158
\(726\) 5.52674e9 0.536032
\(727\) 1.04981e10 1.01330 0.506652 0.862150i \(-0.330883\pi\)
0.506652 + 0.862150i \(0.330883\pi\)
\(728\) −6.38208e7 −0.00613060
\(729\) 3.87420e8 0.0370370
\(730\) −4.09803e9 −0.389893
\(731\) −1.05277e9 −0.0996833
\(732\) −1.20041e9 −0.113120
\(733\) −1.91930e10 −1.80003 −0.900014 0.435862i \(-0.856444\pi\)
−0.900014 + 0.435862i \(0.856444\pi\)
\(734\) −1.92426e10 −1.79608
\(735\) 7.46593e9 0.693551
\(736\) −5.77179e9 −0.533628
\(737\) −6.23235e8 −0.0573477
\(738\) −5.34733e8 −0.0489711
\(739\) −1.50672e10 −1.37334 −0.686669 0.726970i \(-0.740928\pi\)
−0.686669 + 0.726970i \(0.740928\pi\)
\(740\) 4.98539e9 0.452260
\(741\) −2.13531e8 −0.0192796
\(742\) −3.09045e9 −0.277720
\(743\) −1.35355e10 −1.21064 −0.605318 0.795984i \(-0.706954\pi\)
−0.605318 + 0.795984i \(0.706954\pi\)
\(744\) −4.06608e9 −0.361969
\(745\) −1.80707e10 −1.60114
\(746\) 7.07975e9 0.624355
\(747\) 2.47375e9 0.217137
\(748\) 5.34907e8 0.0467329
\(749\) 1.86201e9 0.161918
\(750\) 1.11385e9 0.0964082
\(751\) −4.44209e9 −0.382690 −0.191345 0.981523i \(-0.561285\pi\)
−0.191345 + 0.981523i \(0.561285\pi\)
\(752\) −2.61995e8 −0.0224662
\(753\) −9.19061e9 −0.784444
\(754\) −8.12793e7 −0.00690527
\(755\) −1.56455e10 −1.32305
\(756\) −2.64652e8 −0.0222766
\(757\) −1.96167e10 −1.64358 −0.821789 0.569792i \(-0.807024\pi\)
−0.821789 + 0.569792i \(0.807024\pi\)
\(758\) −1.35186e10 −1.12743
\(759\) 2.88774e9 0.239724
\(760\) 2.62755e10 2.17121
\(761\) −8.13676e9 −0.669276 −0.334638 0.942347i \(-0.608614\pi\)
−0.334638 + 0.942347i \(0.608614\pi\)
\(762\) 8.65837e9 0.708914
\(763\) −4.16005e9 −0.339049
\(764\) 4.20762e9 0.341357
\(765\) 2.36256e9 0.190795
\(766\) 1.33506e9 0.107325
\(767\) 2.95616e7 0.00236562
\(768\) 5.64820e9 0.449931
\(769\) −1.79213e10 −1.42111 −0.710554 0.703642i \(-0.751556\pi\)
−0.710554 + 0.703642i \(0.751556\pi\)
\(770\) 3.64151e9 0.287451
\(771\) 4.31508e9 0.339077
\(772\) −6.78444e9 −0.530705
\(773\) 1.39926e10 1.08961 0.544804 0.838563i \(-0.316604\pi\)
0.544804 + 0.838563i \(0.316604\pi\)
\(774\) −1.22811e9 −0.0952017
\(775\) −1.09845e10 −0.847669
\(776\) 9.12799e8 0.0701228
\(777\) 3.48741e9 0.266704
\(778\) 1.95023e10 1.48476
\(779\) −3.14916e9 −0.238679
\(780\) −5.63578e7 −0.00425229
\(781\) −8.23950e9 −0.618903
\(782\) −5.85740e9 −0.438007
\(783\) 8.68480e8 0.0646537
\(784\) 1.34320e10 0.995485
\(785\) −2.73635e10 −2.01896
\(786\) 4.46174e9 0.327737
\(787\) 7.87847e9 0.576143 0.288071 0.957609i \(-0.406986\pi\)
0.288071 + 0.957609i \(0.406986\pi\)
\(788\) 2.21617e9 0.161347
\(789\) −2.08827e9 −0.151362
\(790\) 9.36468e9 0.675769
\(791\) −3.77601e9 −0.271278
\(792\) −1.60787e9 −0.115004
\(793\) −1.78818e8 −0.0127337
\(794\) 3.55298e9 0.251896
\(795\) 7.03202e9 0.496358
\(796\) 6.02922e9 0.423707
\(797\) −7.12267e9 −0.498355 −0.249178 0.968458i \(-0.580160\pi\)
−0.249178 + 0.968458i \(0.580160\pi\)
\(798\) −7.13325e9 −0.496909
\(799\) −1.06449e8 −0.00738291
\(800\) 8.68179e9 0.599507
\(801\) 6.84749e9 0.470779
\(802\) −4.34517e9 −0.297438
\(803\) 1.47684e9 0.100653
\(804\) 3.22218e8 0.0218652
\(805\) −8.71273e9 −0.588666
\(806\) 2.35067e8 0.0158132
\(807\) −1.20858e10 −0.809503
\(808\) −7.72171e8 −0.0514960
\(809\) −1.13613e10 −0.754412 −0.377206 0.926129i \(-0.623115\pi\)
−0.377206 + 0.926129i \(0.623115\pi\)
\(810\) 2.75605e9 0.182218
\(811\) −1.58411e10 −1.04283 −0.521413 0.853304i \(-0.674595\pi\)
−0.521413 + 0.853304i \(0.674595\pi\)
\(812\) −5.93269e8 −0.0388871
\(813\) −9.16565e9 −0.598200
\(814\) −8.22265e9 −0.534351
\(815\) 1.51311e10 0.979079
\(816\) 4.25050e9 0.273857
\(817\) −7.23263e9 −0.464001
\(818\) −2.86332e10 −1.82908
\(819\) −3.94238e7 −0.00250764
\(820\) −8.31166e8 −0.0526429
\(821\) 2.50210e10 1.57799 0.788993 0.614402i \(-0.210603\pi\)
0.788993 + 0.614402i \(0.210603\pi\)
\(822\) −1.11092e9 −0.0697640
\(823\) 1.51409e10 0.946790 0.473395 0.880850i \(-0.343028\pi\)
0.473395 + 0.880850i \(0.343028\pi\)
\(824\) 6.32329e9 0.393730
\(825\) −4.34366e9 −0.269319
\(826\) 9.87539e8 0.0609711
\(827\) −7.61883e9 −0.468402 −0.234201 0.972188i \(-0.575247\pi\)
−0.234201 + 0.972188i \(0.575247\pi\)
\(828\) −1.49299e9 −0.0914007
\(829\) −1.04307e10 −0.635879 −0.317940 0.948111i \(-0.602991\pi\)
−0.317940 + 0.948111i \(0.602991\pi\)
\(830\) 1.75979e10 1.06829
\(831\) −8.92011e9 −0.539221
\(832\) 1.76868e8 0.0106468
\(833\) 5.45745e9 0.327139
\(834\) −4.81429e9 −0.287377
\(835\) −1.65624e10 −0.984514
\(836\) 3.67486e9 0.217530
\(837\) −2.51172e9 −0.148058
\(838\) −2.20617e10 −1.29505
\(839\) −8.81723e9 −0.515425 −0.257712 0.966222i \(-0.582969\pi\)
−0.257712 + 0.966222i \(0.582969\pi\)
\(840\) 4.85118e9 0.282403
\(841\) −1.53030e10 −0.887137
\(842\) −3.03804e10 −1.75388
\(843\) −1.51123e9 −0.0868829
\(844\) 6.80159e8 0.0389414
\(845\) 2.54187e10 1.44929
\(846\) −1.24178e8 −0.00705098
\(847\) 6.00930e9 0.339807
\(848\) 1.26514e10 0.712445
\(849\) −9.25664e9 −0.519130
\(850\) 8.81056e9 0.492082
\(851\) 1.96736e10 1.09429
\(852\) 4.25990e9 0.235972
\(853\) 6.88811e9 0.379995 0.189998 0.981785i \(-0.439152\pi\)
0.189998 + 0.981785i \(0.439152\pi\)
\(854\) −5.97362e9 −0.328197
\(855\) 1.62310e10 0.888105
\(856\) −5.84866e9 −0.318712
\(857\) 7.74166e9 0.420147 0.210074 0.977686i \(-0.432630\pi\)
0.210074 + 0.977686i \(0.432630\pi\)
\(858\) 9.29537e7 0.00502413
\(859\) −7.99776e9 −0.430519 −0.215259 0.976557i \(-0.569060\pi\)
−0.215259 + 0.976557i \(0.569060\pi\)
\(860\) −1.90893e9 −0.102340
\(861\) −5.81423e8 −0.0310442
\(862\) 3.74514e10 1.99156
\(863\) −2.16260e10 −1.14535 −0.572675 0.819782i \(-0.694094\pi\)
−0.572675 + 0.819782i \(0.694094\pi\)
\(864\) 1.98518e9 0.104713
\(865\) −1.59174e10 −0.836212
\(866\) 8.74875e9 0.457755
\(867\) −9.35216e9 −0.487355
\(868\) 1.71579e9 0.0890523
\(869\) −3.37483e9 −0.174454
\(870\) 6.17824e9 0.318088
\(871\) 4.79991e7 0.00246133
\(872\) 1.30669e10 0.667366
\(873\) 5.63859e8 0.0286828
\(874\) −4.02409e10 −2.03882
\(875\) 1.21111e9 0.0611160
\(876\) −7.63539e8 −0.0383766
\(877\) −2.67027e10 −1.33677 −0.668385 0.743816i \(-0.733014\pi\)
−0.668385 + 0.743816i \(0.733014\pi\)
\(878\) 1.23333e10 0.614961
\(879\) 1.57090e10 0.780167
\(880\) −1.49073e10 −0.737409
\(881\) −1.15396e10 −0.568559 −0.284279 0.958741i \(-0.591754\pi\)
−0.284279 + 0.958741i \(0.591754\pi\)
\(882\) 6.36641e9 0.312431
\(883\) −3.23054e10 −1.57911 −0.789555 0.613679i \(-0.789689\pi\)
−0.789555 + 0.613679i \(0.789689\pi\)
\(884\) −4.11964e7 −0.00200575
\(885\) −2.24705e9 −0.108971
\(886\) 1.57280e10 0.759724
\(887\) 1.57129e10 0.756005 0.378002 0.925805i \(-0.376611\pi\)
0.378002 + 0.925805i \(0.376611\pi\)
\(888\) −1.09541e10 −0.524967
\(889\) 9.41436e9 0.449402
\(890\) 4.87120e10 2.31617
\(891\) −9.93221e8 −0.0470407
\(892\) −3.41478e9 −0.161096
\(893\) −7.31314e8 −0.0343656
\(894\) −1.54094e10 −0.721282
\(895\) 1.98244e10 0.924312
\(896\) 1.07589e10 0.499676
\(897\) −2.22402e8 −0.0102888
\(898\) −3.67670e10 −1.69430
\(899\) −5.63053e9 −0.258458
\(900\) 2.24571e9 0.102685
\(901\) 5.14026e9 0.234125
\(902\) 1.37088e9 0.0621982
\(903\) −1.33534e9 −0.0603512
\(904\) 1.18606e10 0.533970
\(905\) 3.63838e10 1.63169
\(906\) −1.33414e10 −0.596008
\(907\) −6.98597e9 −0.310886 −0.155443 0.987845i \(-0.549681\pi\)
−0.155443 + 0.987845i \(0.549681\pi\)
\(908\) 4.45905e8 0.0197671
\(909\) −4.76990e8 −0.0210637
\(910\) −2.80455e8 −0.0123372
\(911\) −4.32364e9 −0.189468 −0.0947339 0.995503i \(-0.530200\pi\)
−0.0947339 + 0.995503i \(0.530200\pi\)
\(912\) 2.92014e10 1.27474
\(913\) −6.34189e9 −0.275785
\(914\) −6.83958e9 −0.296290
\(915\) 1.35924e10 0.586573
\(916\) −5.56779e9 −0.239358
\(917\) 4.85131e9 0.207762
\(918\) 2.01462e9 0.0859496
\(919\) −2.41430e10 −1.02609 −0.513046 0.858361i \(-0.671483\pi\)
−0.513046 + 0.858361i \(0.671483\pi\)
\(920\) 2.73670e10 1.15870
\(921\) −1.67891e10 −0.708138
\(922\) 3.62437e10 1.52291
\(923\) 6.34574e8 0.0265630
\(924\) 6.78482e8 0.0282935
\(925\) −2.95926e10 −1.22938
\(926\) 3.74441e10 1.54969
\(927\) 3.90606e9 0.161050
\(928\) 4.45017e9 0.182792
\(929\) 2.15464e10 0.881699 0.440850 0.897581i \(-0.354677\pi\)
0.440850 + 0.897581i \(0.354677\pi\)
\(930\) −1.78680e10 −0.728428
\(931\) 3.74932e10 1.52275
\(932\) −5.70135e9 −0.230686
\(933\) −2.12222e10 −0.855471
\(934\) 1.56738e10 0.629449
\(935\) −6.05684e9 −0.242329
\(936\) 1.23832e8 0.00493590
\(937\) 3.83047e10 1.52112 0.760561 0.649267i \(-0.224924\pi\)
0.760561 + 0.649267i \(0.224924\pi\)
\(938\) 1.60346e9 0.0634380
\(939\) −1.00616e10 −0.396584
\(940\) −1.93018e8 −0.00757966
\(941\) −2.56605e10 −1.00392 −0.501962 0.864890i \(-0.667388\pi\)
−0.501962 + 0.864890i \(0.667388\pi\)
\(942\) −2.33336e10 −0.909502
\(943\) −3.27999e9 −0.127374
\(944\) −4.04269e9 −0.156411
\(945\) 2.99670e9 0.115513
\(946\) 3.14848e9 0.120916
\(947\) 4.39924e10 1.68326 0.841632 0.540051i \(-0.181595\pi\)
0.841632 + 0.540051i \(0.181595\pi\)
\(948\) 1.74481e9 0.0665151
\(949\) −1.13740e8 −0.00431999
\(950\) 6.05294e10 2.29052
\(951\) 6.42967e9 0.242414
\(952\) 3.54611e9 0.133206
\(953\) −4.24635e8 −0.0158924 −0.00794622 0.999968i \(-0.502529\pi\)
−0.00794622 + 0.999968i \(0.502529\pi\)
\(954\) 5.99640e9 0.223600
\(955\) −4.76435e10 −1.77008
\(956\) 1.02164e10 0.378176
\(957\) −2.22650e9 −0.0821166
\(958\) 2.20473e10 0.810170
\(959\) −1.20792e9 −0.0442255
\(960\) −1.34442e10 −0.490439
\(961\) −1.12286e10 −0.408126
\(962\) 6.33276e8 0.0229340
\(963\) −3.61287e9 −0.130365
\(964\) 7.45127e9 0.267893
\(965\) 7.68213e10 2.75192
\(966\) −7.42959e9 −0.265182
\(967\) 4.76260e10 1.69376 0.846879 0.531786i \(-0.178479\pi\)
0.846879 + 0.531786i \(0.178479\pi\)
\(968\) −1.88755e10 −0.668857
\(969\) 1.18646e10 0.418908
\(970\) 4.01122e9 0.141116
\(971\) −1.01771e10 −0.356744 −0.178372 0.983963i \(-0.557083\pi\)
−0.178372 + 0.983963i \(0.557083\pi\)
\(972\) 5.13504e8 0.0179354
\(973\) −5.23465e9 −0.182177
\(974\) −3.05948e10 −1.06094
\(975\) 3.34532e8 0.0115590
\(976\) 2.44542e10 0.841936
\(977\) 5.07713e10 1.74176 0.870878 0.491499i \(-0.163551\pi\)
0.870878 + 0.491499i \(0.163551\pi\)
\(978\) 1.29027e10 0.441056
\(979\) −1.75547e10 −0.597937
\(980\) 9.89567e9 0.335857
\(981\) 8.07174e9 0.272977
\(982\) −6.55867e9 −0.221017
\(983\) 3.69088e10 1.23934 0.619672 0.784861i \(-0.287266\pi\)
0.619672 + 0.784861i \(0.287266\pi\)
\(984\) 1.82627e9 0.0611058
\(985\) −2.50941e10 −0.836651
\(986\) 4.51617e9 0.150038
\(987\) −1.35021e8 −0.00446983
\(988\) −2.83024e8 −0.00933627
\(989\) −7.53310e9 −0.247621
\(990\) −7.06563e9 −0.231434
\(991\) 5.58119e10 1.82167 0.910833 0.412774i \(-0.135440\pi\)
0.910833 + 0.412774i \(0.135440\pi\)
\(992\) −1.28703e10 −0.418598
\(993\) 1.75494e10 0.568775
\(994\) 2.11987e10 0.684630
\(995\) −6.82699e10 −2.19709
\(996\) 3.27882e9 0.105150
\(997\) −4.81978e9 −0.154026 −0.0770130 0.997030i \(-0.524538\pi\)
−0.0770130 + 0.997030i \(0.524538\pi\)
\(998\) −4.63036e10 −1.47454
\(999\) −6.76663e9 −0.214730
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.d.1.5 18
3.2 odd 2 531.8.a.e.1.14 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.5 18 1.1 even 1 trivial
531.8.a.e.1.14 18 3.2 odd 2